Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d)$H^s\left({\mathbb{T}}^d\right)$ of fractional smoothness s>0$s>0$ for various equivalent norms including the classical one. The error is always measured in _L2(T^d)$L_2\left({\mathbb{T}}^d\right)$. Particular emphasis is given to the dependence of all constants on the dimension d$d$. We capture the exact decay rate in n$n$ and the exact decay order of the constants with respect to d$d$, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.     

Landau常数的逼近公式

对于所有的整数n≥0,Landau常数定义为Gn=nΣk=01/16k(2kk)2.该文建立了Landau常数新的逼近公式.指出了获得的结果与先前已有结果之间的相关联系     

On the pion decay constant

The pion decay constant fπ$f_{\pi }$ plays a crucial role in many areas of low energy particle physics. Its value may e.g. be deduced from experimental data on leptonic pion decays. Here, we provide comments on several aspects of this evaluation. In particular, we point out that at the present level of experimental accuracy, the value of fπ$f_{\pi }$ is sensitive to the value of the pion mass chosen in its chiral expansion.     

采用中子衍射实验解析PbS的原子间力常数

使用日本原子能研究开发机构里设置的高分解能粉末中子衍射装置, 在 15 K 和 294 K 温度下对 PbS 进行了中子衍射实验。在 15 K 和 294 K 温度里观察到了明显的振动形状的热漫散射。 所观察到的漫散射强度可由原子间热振动相关效应的影响来解释。在 294 K 下从原子间相关效应和德拜—劳厄因子的温度参数中获得了 PbS 相邻原子间的力常数。     

Structural, elastic, and electronic properties of orthorhombic NH3BH3: An ab initio study

At the generalized gradient approximation (GGA), the elastic constants of the orthorhombic phase of NH₃BH₃ were calculated with plane-wave pseudo-potential method. Our calculation showed that the orthorhombic phase NH₃BH₃ is a loose and brittle material, as well as hard to be deformed, also we calculated the elastic anisotropies and the Debye temperatures from the elastic constants. And from the band structure and density of state (DOS), we concluded that NH₃BH₃ is a wide-gap semiconductor and the band gap is almost 6.0 eV. The bonds between N atoms and H atoms show a strong covalent characteristic, B atoms and H atoms form ironic bonds, and so as to the B-N bonds. Electrons from the B atoms are absorbed by the H atoms around the B atoms, and the H atoms display electronegativity.     

1-Ethoxycarbonyl-3-ferrocenyl-propan-1,3-dion und Ferrocen-1,1'bis(2,4-dioxobutansäureethylester) als Liganden für Übergangsmetallionen. Kristallstruktur von Bis(1-ethoxycarbonyl-3-ferrocenyl-propan-1,3dionato)kupfer(II)

1-Ethoxycarbonyl-3-ferrocenyl-propane-1,3-dion and Ferrocene-1,1′bis(2,4-dioxobutanoic acid ethylester) as Ligands for Transition Metal Ions. Crystal Structure of Bis(1-ethoxycarbonyl-3-ferrocenyl-propane-1,3dionato)copper(II) The ligands 1-ethoxycarbonyl-3-ferrocenyl-propane-1,3-dion and ferrocene-1,1′-bis(2,4-dioxo-butanoic acid ethylester) have been prepared by reaction of acetylferrocene or 1,1′-diacetylferrocene and diethyl oxalate. They yield neutral chelates with Cu^II, Ni^II, Zn^II, Co^II, and Mn^II. The acid dissociation constants of the ligands and the stability constants of their metal complexes including Fe^II complexes are reported. The structure of bis(1-ethoxycarbonyl-3-ferrocenyl-propane-1,3-dionato)copper(II) was determined by X-ray structure analysis. A cis arrangement with a nearly square planar coordination sphere at the Cu atom is found.     

Hydrolysis of (CH3)3Sn+ in Various Salt Media

The hydrolysis of trimethyltin(IV) has been studied by potentiometry (H⁺ -glass electrode) and calorimetry in various salt media (NaNO₃, NaCl, KCl, Na₂SO₄, and NaNO₃—NaCl mixtures). The effect of ionic strength on the hydrolysis constants is accounted for by a simple Debye–Hückel type equation and by Pitzer equations. The results allow us to obtain H for hydrolysis and the temperature dependence of the Pitzer parameters. The resulting coefficients can be used to examine the speciation of (CH₃)₃Sn⁺ in multicomponent electrolyte solutions, such as natural waters, over a wide range of temperature and ionic strength.     

Salting-out gradients in centrifugal partition chromatography for the isolation of chlorogenic acids from green coffee beans

In addition to sample solubility constraints, the use of polarity gradients in normal-phase centrifugal partition chromatography (CPC) for the purification of complex mixtures is also limited by the instability of biphasic systems as a consequence of dramatic changes in the settling times along the gradient, leading in many cases to column bleeding when working under maximum efficiency conditions. In this paper an electrostriction approach is proposed as a strategy in reversed-phase CPC to fractionate intermediate polarity extracts in a single run by bringing its components into the “sweet spot” in a controlled fashion through a stepwise reduction of salt concentration in the aqueous mobile phase. The salting-out gradient method was successfully tested with the separation of the major chlorogenic acids (CGAs, hydroxycinnamoylquinic acids) present in green coffee beans (5-caffeoylquinic acid (5-CQA), 5-feruloylquinic acid (5-FQA) and 3,5-dicaffeoylquinic acid (3,5-diCQA)) using ethyl acetate-hexane as the stationary phase and an ionic gradient of LiCl (5.0, 2.5 and 0.1 M) as the mobile phase in one case and (NH₄)₂SO₄/KNO₃ (3.0 and 1.5 M/1.5 M) in another. Regioisomers of each chlorogenic acid obtained by base-catalyzed isomerisation were also separated by CPC using isocratic elution. The best resolution for both FQAs and diCQAs was achieved with a chloroform–n-butanol–0.01 M pH 2.5 phosphate buffer (84:16:100; v/v) system, while CQAs were best isolated using chloroform–n-butanol–0.01 M pH 2.5 phosphate buffer/5.0 M LiCl (82:18:100; v/v).     

Raman spectral study of the constitution and equilibria of nitric acid and d-nitric acid

From relative integrated intensity measurements of the symmetric stretching vibration of nitrate ion in nitric acid solutions (both HNO ₃ /H ₂ O and DNO ₃ /D ₂ O), the mass law concentration quotients, Q were obtained as functions of concentration. By extrapolation the limiting dissociation constants were estimated to be 24.4 and 15 respectively at 25°C. It is shown that this constant refers to a process in which the ion pair H ₃ O⁺ NO ₃ ^– is in equilibrium with the dispersed, solvated H ₃ O⁺ and NO ₃ ^– ions.     

Theoretical study on the reaction CX3 + SiH(CH3)3 (X = H, F)

Theoretical investigations are carried out on the multiple-channel reactions, CH(3) + SiH(CH(3))(3) → products and CF(3) + SiH(CH(3))(3) → products. The minimum energy paths (MEP) are calculated at the MP2/6-311 + G(d,p) level, and energetic information is further refined by the MC-QCISD (single point) method. The rate constants for major reaction channels are calculated by the canonical variational transition state theory (CVT) with small-curvature tunneling (SCT) correction over the temperature range 200-1500 K. The theoretical rate constants are in good agreement with the available experimental data and are found to be k(1a)(T) = 1.93 × 10(-24) T(3.15) exp(-1214.59/T) and k(2a)(T) = 1.33 × 10(-25) T(4.13) exp(-397.94/T) (in unit of cm(3) molecule(-1) s(-1)). Our calculations indicate that hydrogen abstraction channel from SiH group is the major channel due to the smaller barrier height among five channels considered.